AN EFFICIENT APPROACH FOR STATISTICAL CALCULATIONS WITH GLOBALLY
GRIDDED FILTERED TIME SERIES

J. Climate Res., 1, 429-434 (1988)

Introduction

In Atmospheric observational studies many statistical concepts and
calculations are involved, for example, variance, spectrum, correlation,
EOF (Empirical Orthogonal Function), EEOF (Extended Empirical Orthogonal
Function), etc. Some statistical calculations are quite simple, while
others may be lengthy and expensive. In many cases, use of the
frequency domain can result in improved efficiency for statistical
calculations with time-filtered data. Only the Fourier coefficients
within the filtered frequency band are significantly nonzero; moreover,
these coefficients contain all the information of interest. This paper
discusses several statistical calculations based on Fourier coefficients
and presents the formulas needed to effect these calculations:

1) A method is given which allows an estimate of the degrees of
temporal freedom of two correlated time series, utilizing the frequency
spectra of these two time series.

2) A simple way to perform seasonal analyses is proposed which employs
a half-year summer/winter projection operator in the frequency domain.

3) A modified lag-correlation calculation is suggested from which lag
correlations may be easily obtained in the frequency domain.

4) A spectral approach is presented for EOF and EEOF analyses which
reduces the size of the matrix to be solved in the eigen problem. No
matter how many grid points are covered by the analyzed area or how many
lag steps are involved, the size of the matrix to be solved is always
s x s where s is the number of Fourier coefficients in the filter
bandpass, resulting in a significant reduction in computation time.

Some of these calculation methods have been used in the study of low
frequency oscillations in the large-scale stratospheric temperature field
(Gao and Stanford 1988).